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	<title>Undergrad math not in mathlib</title>
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                <h1>Missing undergraduate mathematics in mathlib</h1>

                <p>
				This gives pointers to undergraduate maths topics that are
				currently missing in mathlib.
				There is also a page listing undergraduate maths topics
				that are 
				<a href="undergrad.html">already in mathlib</a>.
				</p>

				<p>If you want to work on an item from this list then
				you should first check the 
				<a href="https://github.com/leanprover-community/mathlib/pulls">pull requests list</a>
				to see whether it is already coming, then the 
				<a href="https://github.com/leanprover-community/mathlib/issues">issues list</a>
				to see whether it is discussed there, and finally talk
				about this idea on <a href="https://leanprover.zulipchat.com/">Zulip</a>.
				<p/>

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                                transpose of a linear map, 
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                                orthogonality
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                                                        Finite-dimensional vector spaces
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                                rank of a set of vectors, 
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                                rank of a system of linear equations, 
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                                                        Multilinearity
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                                determinant of vectors, 
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                                determinant of endomorphisms, 
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                                special linear group, 
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                                orientation of a $\R$-valued vector space
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                                                        Matrices
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                                change of basis, 
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                                rank of a matrix, 
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                                elementary row operations, 
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                                Gaussian elimination, 
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                                row-reduced matrices
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                                                        Endomorphism polynomials
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                                annihilating polynomials, 
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                                                        Structure theory of endomorphisms
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                                eigenvalues, 
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                                eigenvectors, 
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                                diagonalization, 
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                                triangularization, 
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                                invariant subspaces of an endomorphism, 
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                                generalized eigenspaces, 
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                                kernels lemma, 
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                                Dunford decomposition, 
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                                Jordan normal form
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                                                        Linear representations
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                                irreducible representation, 
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                                Schur's lemma, 
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                                                        Exponential
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                                endomorphism exponential, 
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                                matrix exponential
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                                                        Group Theory
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                                        </h5>
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                                                        Basic definitions
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                                class formula, 
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                                conjugacy classes
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                                                        Abelian group
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                                                <a class="hover-link" href="#1-1">#</a>
                                        </h5>
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                                finite type abelian groups, 
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                                complex roots of unity, 
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                                primitive complex roots of unity
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                                                        Permutation group
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                                decomposition into transpositions, 
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                                decomposition into cycles with disjoint support, 
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                                alternating group
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                                                        Classical automorphism groups
                                                </a>
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                                orthogonal group, 
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                                special orthogonal group, 
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                                unitary group, 
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                                special unitary group
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                                                        Representation theory of finite groups
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                                representations of abelian groups, 
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                                dual groups, 
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                                Maschke theorem, 
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                                orthogonality of irreducible characters, 
                        </span>
                        
                        
                        
						
                        
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                                Fourier transform for finite abelian groups, 
                        </span>
                        
                        
                        
						
                        
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                                convolution, 
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                                class function over a group, 
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                                characters of a finite dimensional representation, 
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                                orthonormal basis of irreducible characters, 
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                                examples of groups with small cardinality
                        </span>
                        
                        
                        
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                                                        Ring Theory
                                                </a>
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                                        </h5>
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                                        <h5 class="card-title markdown-heading" id="2-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-2-2" aria-expanded="false" aria-controls="collapse-2-2">
                                                        Algebra
                                                </a>
                                                <a class="hover-link" href="#2-2">#</a>
                                        </h5>
                                        <div id="collapse-2-2" class="collapse" data-parent="#overview-accordion-2 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-2-2" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                associative algebra over a commutative ring
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-2-3">
                                        <h5 class="card-title markdown-heading" id="2-3">
                                                <a role="button" data-toggle="collapse" href="#collapse-2-3" aria-expanded="false" aria-controls="collapse-2-3">
                                                        Divisibility in integral domains
                                                </a>
                                                <a class="hover-link" href="#2-3">#</a>
                                        </h5>
                                        <div id="collapse-2-3" class="collapse" data-parent="#overview-accordion-2 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-2-3" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $A[X]$ is a UFD when $A$ is a UFD, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Bezout's identity, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-2-4">
                                        <h5 class="card-title markdown-heading" id="2-4">
                                                <a role="button" data-toggle="collapse" href="#collapse-2-4" aria-expanded="false" aria-controls="collapse-2-4">
                                                        Polynomial rings
                                                </a>
                                                <a class="hover-link" href="#2-4">#</a>
                                        </h5>
                                        <div id="collapse-2-4" class="collapse" data-parent="#overview-accordion-2 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-2-4" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                cyclotomic polynomials in $\Q[X]$, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                relationship between the coefficients and the roots of a split polynomial, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Newton's identities, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                decomposition into sums of homogeneous polynomials, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                symmetric polynomials
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-2-5">
                                        <h5 class="card-title markdown-heading" id="2-5">
                                                <a role="button" data-toggle="collapse" href="#collapse-2-5" aria-expanded="false" aria-controls="collapse-2-5">
                                                        Field Theory
                                                </a>
                                                <a class="hover-link" href="#2-5">#</a>
                                        </h5>
                                        <div id="collapse-2-5" class="collapse" data-parent="#overview-accordion-2 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-2-5" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                transcendental elements, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                algebraically closed fields, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                splitting fields, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                rational fraction fields with one indeterminate over a field, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $\R(X)$-partial fraction decomposition, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $\C(X)$-partial fraction decomposition
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-3">
                                        <h5 class="card-title markdown-heading" id="3">
                                                <a role="button" data-toggle="collapse" href="#collapse-3" aria-expanded="false" aria-controls="collapse-3">
                                                        Bilinear and Quadratic Forms Over a Vector Space
                                                </a>
                                                <a class="hover-link" href="#3">#</a>
                                        </h5>
                                        <div id="collapse-3" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-3" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-3-0">
                                        <h5 class="card-title markdown-heading" id="3-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-3-0" aria-expanded="false" aria-controls="collapse-3-0">
                                                        Bilinear forms
                                                </a>
                                                <a class="hover-link" href="#3-0">#</a>
                                        </h5>
                                        <div id="collapse-3-0" class="collapse" data-parent="#overview-accordion-3 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-3-0" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                nondegenerate forms, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                rank of a bilinear form
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-3-2">
                                        <h5 class="card-title markdown-heading" id="3-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-3-2" aria-expanded="false" aria-controls="collapse-3-2">
                                                        Orthogonality
                                                </a>
                                                <a class="hover-link" href="#3-2">#</a>
                                        </h5>
                                        <div id="collapse-3-2" class="collapse" data-parent="#overview-accordion-3 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-3-2" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                adjoint endomorphism, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Sylvester's law of inertia, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                real classification, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                complex classification, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Gram-Schmidt orthogonalisation
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-3-3">
                                        <h5 class="card-title markdown-heading" id="3-3">
                                                <a role="button" data-toggle="collapse" href="#collapse-3-3" aria-expanded="false" aria-controls="collapse-3-3">
                                                        Euclidean and Hermitian spaces
                                                </a>
                                                <a class="hover-link" href="#3-3">#</a>
                                        </h5>
                                        <div id="collapse-3-3" class="collapse" data-parent="#overview-accordion-3 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-3-3" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Hermitian vector spaces, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                dual isomorphism in the euclidean case, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                orthogonal complement, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                orthonormal bases
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-3-4">
                                        <h5 class="card-title markdown-heading" id="3-4">
                                                <a role="button" data-toggle="collapse" href="#collapse-3-4" aria-expanded="false" aria-controls="collapse-3-4">
                                                        Endomorphisms
                                                </a>
                                                <a class="hover-link" href="#3-4">#</a>
                                        </h5>
                                        <div id="collapse-3-4" class="collapse" data-parent="#overview-accordion-3 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-3-4" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                orthogonal group, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                unitary group, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                special orthogonal group, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                special unitary group, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                normal endomorphism, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                diagonalization of a self-adjoint endomorphism, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                diagonalization of normal endomorphisms, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                simultaneous diagonalization of two real quadratic forms, with one positive-definite, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                decomposition of an orthogonal transformation as a product of reflections, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                polar decompositions in $\mathrm{GL}(n, \R)$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                polar decompositions in $\mathrm{GL}(n, \C)$
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-3-5">
                                        <h5 class="card-title markdown-heading" id="3-5">
                                                <a role="button" data-toggle="collapse" href="#collapse-3-5" aria-expanded="false" aria-controls="collapse-3-5">
                                                        Low dimensions
                                                </a>
                                                <a class="hover-link" href="#3-5">#</a>
                                        </h5>
                                        <div id="collapse-3-5" class="collapse" data-parent="#overview-accordion-3 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-3-5" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                cross product, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                triple product, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                classification of elements of $\mathrm{O}(2, \R)$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                classification of elements of $\mathrm{O}(3, \R)$
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-4">
                                        <h5 class="card-title markdown-heading" id="4">
                                                <a role="button" data-toggle="collapse" href="#collapse-4" aria-expanded="false" aria-controls="collapse-4">
                                                        Affine and Euclidian Geometry
                                                </a>
                                                <a class="hover-link" href="#4">#</a>
                                        </h5>
                                        <div id="collapse-4" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-4" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-4-0">
                                        <h5 class="card-title markdown-heading" id="4-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-4-0" aria-expanded="false" aria-controls="collapse-4-0">
                                                        General definitions
                                                </a>
                                                <a class="hover-link" href="#4-0">#</a>
                                        </h5>
                                        <div id="collapse-4-0" class="collapse" data-parent="#overview-accordion-4 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-4-0" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                equations of affine subspace, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                affine groups, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                affine property, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                homothetic transformation groups, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                affinity
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-4-1">
                                        <h5 class="card-title markdown-heading" id="4-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-4-1" aria-expanded="false" aria-controls="collapse-4-1">
                                                        Convexity
                                                </a>
                                                <a class="hover-link" href="#4-1">#</a>
                                        </h5>
                                        <div id="collapse-4-1" class="collapse" data-parent="#overview-accordion-4 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-4-1" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                convex subsets, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                convex hull of a subset of an affine real space, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                extreme point
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-4-2">
                                        <h5 class="card-title markdown-heading" id="4-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-4-2" aria-expanded="false" aria-controls="collapse-4-2">
                                                        Euclidean affine spaces
                                                </a>
                                                <a class="hover-link" href="#4-2">#</a>
                                        </h5>
                                        <div id="collapse-4-2" class="collapse" data-parent="#overview-accordion-4 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-4-2" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                isometries of a Euclidian affine subspace, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Euclidian affine space isometry group, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Euclidean affine space isometries, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                isometries that do and do not preserve orientation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                direct and indirect similarities of the plane, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                isometric classification in two and three dimensions, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                angles formed by planes, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                inscribed angle theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                cocyclicity, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                group of isometries stabilizing subset of the plane or space, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                regular polygons, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                metric relations in the triangle, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                using complex numbers in plane geometry
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-4-3">
                                        <h5 class="card-title markdown-heading" id="4-3">
                                                <a role="button" data-toggle="collapse" href="#collapse-4-3" aria-expanded="false" aria-controls="collapse-4-3">
                                                        Application of quadratic forms to study proper conic sections of the affine euclidean plane
                                                </a>
                                                <a class="hover-link" href="#4-3">#</a>
                                        </h5>
                                        <div id="collapse-4-3" class="collapse" data-parent="#overview-accordion-4 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-4-3" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                focus, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                eccentricity, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                quadratics on 3 dimensional euclidean affine spaces
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5">
                                        <h5 class="card-title markdown-heading" id="5">
                                                <a role="button" data-toggle="collapse" href="#collapse-5" aria-expanded="false" aria-controls="collapse-5">
                                                        Single Variable Real Analysis
                                                </a>
                                                <a class="hover-link" href="#5">#</a>
                                        </h5>
                                        <div id="collapse-5" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-2">
                                        <h5 class="card-title markdown-heading" id="5-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-2" aria-expanded="false" aria-controls="collapse-5-2">
                                                        Topology of R
                                                </a>
                                                <a class="hover-link" href="#5-2">#</a>
                                        </h5>
                                        <div id="collapse-5-2" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-2" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                additive subgroups of $\R$
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-3">
                                        <h5 class="card-title markdown-heading" id="5-3">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-3" aria-expanded="false" aria-controls="collapse-5-3">
                                                        Numerical Series
                                                </a>
                                                <a class="hover-link" href="#5-3">#</a>
                                        </h5>
                                        <div id="collapse-5-3" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-3" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Convergence of real valued-series, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Riemann series, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                positive valued series, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                summation of comparison relations, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                comparison of a series and an integral, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                error estimation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                absolute convergence, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                products of series, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                alternating series
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-4">
                                        <h5 class="card-title markdown-heading" id="5-4">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-4" aria-expanded="false" aria-controls="collapse-5-4">
                                                        Real-valued functions defined on a subset of $\R$
                                                </a>
                                                <a class="hover-link" href="#5-4">#</a>
                                        </h5>
                                        <div id="collapse-5-4" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-4" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                continuity of monotonic functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                continuity of reciprocal functions
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-5">
                                        <h5 class="card-title markdown-heading" id="5-5">
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                                                        Differentiability
                                                </a>
                                                <a class="hover-link" href="#5-5">#</a>
                                        </h5>
                                        <div id="collapse-5-5" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-5" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                piecewise $C^k$ functions, 
                        </span>
                        
                        
                        
						
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-6">
                                        <h5 class="card-title markdown-heading" id="5-6">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-6" aria-expanded="false" aria-controls="collapse-5-6">
                                                        Taylor-like theorems
                                                </a>
                                                <a class="hover-link" href="#5-6">#</a>
                                        </h5>
                                        <div id="collapse-5-6" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-6" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Taylor's theorem with little-o remainder, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Taylor's theorem with integral form for remainder, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Taylor's theorem with Lagrange form for remainder, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Taylor series expansions
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-7">
                                        <h5 class="card-title markdown-heading" id="5-7">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-7" aria-expanded="false" aria-controls="collapse-5-7">
                                                        Usual functions (trigonometric, rational, $\exp$, $\log$, etc)
                                                </a>
                                                <a class="hover-link" href="#5-7">#</a>
                                        </h5>
                                        <div id="collapse-5-7" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-7" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                rational functions, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                reciprocal hyperbolic trigonometric functions
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-8">
                                        <h5 class="card-title markdown-heading" id="5-8">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-8" aria-expanded="false" aria-controls="collapse-5-8">
                                                        Integration
                                                </a>
                                                <a class="hover-link" href="#5-8">#</a>
                                        </h5>
                                        <div id="collapse-5-8" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-8" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                integral over a segment of piecewise continuous functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                antiderivatives, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Riemann sums, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                antiderivative of a continuous function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                usual methods to compute integrals, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                change of variable, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                integration by parts, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                generalized integrals, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                absolutely convergent integrals, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                integration of asymptotic comparison relationships, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                semi-convergent integrals
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-9">
                                        <h5 class="card-title markdown-heading" id="5-9">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-9" aria-expanded="false" aria-controls="collapse-5-9">
                                                        Sequences and series of functions
                                                </a>
                                                <a class="hover-link" href="#5-9">#</a>
                                        </h5>
                                        <div id="collapse-5-9" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-9" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                pointwise convergence, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                normal convergence, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                differentiability of the limit, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Weierstrass polynomial approximation theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Weierstrass trigonometric approximation theorem
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-5-10">
                                        <h5 class="card-title markdown-heading" id="5-10">
                                                <a role="button" data-toggle="collapse" href="#collapse-5-10" aria-expanded="false" aria-controls="collapse-5-10">
                                                        Convexity
                                                </a>
                                                <a class="hover-link" href="#5-10">#</a>
                                        </h5>
                                        <div id="collapse-5-10" class="collapse" data-parent="#overview-accordion-5 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-5-10" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                continuity and differentiability of convex functions, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-6">
                                        <h5 class="card-title markdown-heading" id="6">
                                                <a role="button" data-toggle="collapse" href="#collapse-6" aria-expanded="false" aria-controls="collapse-6">
                                                        Single Variable Complex Analysis
                                                </a>
                                                <a class="hover-link" href="#6">#</a>
                                        </h5>
                                        <div id="collapse-6" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-6" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-6-0">
                                        <h5 class="card-title markdown-heading" id="6-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-6-0" aria-expanded="false" aria-controls="collapse-6-0">
                                                        Complex Valued series
                                                </a>
                                                <a class="hover-link" href="#6-0">#</a>
                                        </h5>
                                        <div id="collapse-6-0" class="collapse" data-parent="#overview-accordion-6 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-6-0" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                radius of convergence, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                properties of sums of complex valued series on their disks of convergence, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                continuity, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                differentiability with respect to the complex variable, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                antiderivative, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                complex exponentials, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                extension of circular functions to the complex plane, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                power series expansion of usual functions
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-6-1">
                                        <h5 class="card-title markdown-heading" id="6-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-6-1" aria-expanded="false" aria-controls="collapse-6-1">
                                                        Functions on one complex variable
                                                </a>
                                                <a class="hover-link" href="#6-1">#</a>
                                        </h5>
                                        <div id="collapse-6-1" class="collapse" data-parent="#overview-accordion-6 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-6-1" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                holomorphic functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Cauchy-Riemann conditions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                line integrals of continuous functions in $\C$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                antiderivatives of a holomorphic function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                representations of the $\log$ function on $\C$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                theorem of holomorphic functions under integral domains, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                winding number of a closed curve in $\C$ with respect to a point, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Cauchy formulas, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                analyticity of a holomorphic function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                principle of isolated zeros, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                principle of analytic continuation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                maximum principle, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                isolated singularities, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Laurent series, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                meromorphic functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                residue theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                sequences and series of holomorphic functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                holomorphic stability under uniform convergence
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-7">
                                        <h5 class="card-title markdown-heading" id="7">
                                                <a role="button" data-toggle="collapse" href="#collapse-7" aria-expanded="false" aria-controls="collapse-7">
                                                        Topology
                                                </a>
                                                <a class="hover-link" href="#7">#</a>
                                        </h5>
                                        <div id="collapse-7" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-7" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-7-0">
                                        <h5 class="card-title markdown-heading" id="7-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-7-0" aria-expanded="false" aria-controls="collapse-7-0">
                                                        Topology and Metric Spaces
                                                </a>
                                                <a class="hover-link" href="#7-0">#</a>
                                        </h5>
                                        <div id="collapse-7-0" class="collapse" data-parent="#overview-accordion-7 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-7-0" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                path connectedness, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-7-1">
                                        <h5 class="card-title markdown-heading" id="7-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-7-1" aria-expanded="false" aria-controls="collapse-7-1">
                                                        Normed vector spaces on $\R$ and $\C$
                                                </a>
                                                <a class="hover-link" href="#7-1">#</a>
                                        </h5>
                                        <div id="collapse-7-1" class="collapse" data-parent="#overview-accordion-7 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-7-1" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                equivalent norms, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Riesz' lemma (unit-ball characterization of finite dimension), 
                        </span>
                        
                        
                        
						
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-7-2">
                                        <h5 class="card-title markdown-heading" id="7-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-7-2" aria-expanded="false" aria-controls="collapse-7-2">
                                                        Hilbert Spaces
                                                </a>
                                                <a class="hover-link" href="#7-2">#</a>
                                        </h5>
                                        <div id="collapse-7-2" class="collapse" data-parent="#overview-accordion-7 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-7-2" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Hilbert projection theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                orthogonal projection onto closed vector subspaces, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Riesz representation theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $l^2$ and $L^2$ cases, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Hilbert (orthonormal) bases (in the separable case), 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Hilbert basis of trigonometric polynomials, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Hilbert bases of orthogonal polynomials, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Lax-Milgram theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $H^1_0([0,1])$ and its application to the one-dimensional Dirichlet problem
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-8">
                                        <h5 class="card-title markdown-heading" id="8">
                                                <a role="button" data-toggle="collapse" href="#collapse-8" aria-expanded="false" aria-controls="collapse-8">
                                                        Multivariable calculus
                                                </a>
                                                <a class="hover-link" href="#8">#</a>
                                        </h5>
                                        <div id="collapse-8" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-8" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-8-0">
                                        <h5 class="card-title markdown-heading" id="8-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-8-0" aria-expanded="false" aria-controls="collapse-8-0">
                                                        Differential Calculus
                                                </a>
                                                <a class="hover-link" href="#8-0">#</a>
                                        </h5>
                                        <div id="collapse-8-0" class="collapse" data-parent="#overview-accordion-8 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-8-0" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                directional derivative, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                partial derivatives, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Jacobian matrix, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                gradient vector, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Hessian matrix, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $k$-th order partial derivatives, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                partial derivatives commute, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Taylor's theorem with little-o remainder, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Taylor's theorem with integral form for remainder, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-8-1">
                                        <h5 class="card-title markdown-heading" id="8-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-8-1" aria-expanded="false" aria-controls="collapse-8-1">
                                                        Differential equations
                                                </a>
                                                <a class="hover-link" href="#8-1">#</a>
                                        </h5>
                                        <div id="collapse-8-1" class="collapse" data-parent="#overview-accordion-8 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-8-1" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Cauchy-Lipschitz Theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                maximal solutions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Grönwall lemma, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                exit theorem of a compact subspace, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                autonomous differential equations, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                phase portraits, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                qualitative behavior, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                stability of equilibrium points (linearisation theorem), 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                linear differential systems, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                method of constant variation (Duhamel’s formula), 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                constant coefficient case, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                solving systems of differential equations of order $> 1$
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-8-2">
                                        <h5 class="card-title markdown-heading" id="8-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-8-2" aria-expanded="false" aria-controls="collapse-8-2">
                                                        Submanifolds of $\R^n$
                                                </a>
                                                <a class="hover-link" href="#8-2">#</a>
                                        </h5>
                                        <div id="collapse-8-2" class="collapse" data-parent="#overview-accordion-8 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-8-2" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                local graphs, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                local parameterization, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                local equation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                tangent space, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                position with respect to the tangent plane, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                gradient, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                line integral, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                curve length, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Lagrange multipliers
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-9">
                                        <h5 class="card-title markdown-heading" id="9">
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                                                        Measures and integral Calculus
                                                </a>
                                                <a class="hover-link" href="#9">#</a>
                                        </h5>
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                                                <div class="card-body">
                                                        <div id="overview-accordion-9" class="accordion">
                                                                
						
                        
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-9-1">
                                        <h5 class="card-title markdown-heading" id="9-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-9-1" aria-expanded="false" aria-controls="collapse-9-1">
                                                        Integration
                                                </a>
                                                <a class="hover-link" href="#9-1">#</a>
                                        </h5>
                                        <div id="collapse-9-1" class="collapse" data-parent="#overview-accordion-9 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-9-1" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                continuity of integrals with respect to parameters, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                differentiability of integrals with respect to parameters, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $\mathrm{L}^p$ spaces where $1 ≤ p ≤ ∞$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Completeness of $\mathrm{L}^p$ spaces, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Holder's inequality, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Fubini's theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                change of variables for multiple integrals, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                integration by parts, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                convolution, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                regularization and approximation by convolution
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-9-2">
                                        <h5 class="card-title markdown-heading" id="9-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-9-2" aria-expanded="false" aria-controls="collapse-9-2">
                                                        Fourier Analysis
                                                </a>
                                                <a class="hover-link" href="#9-2">#</a>
                                        </h5>
                                        <div id="collapse-9-2" class="collapse" data-parent="#overview-accordion-9 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-9-2" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Fourier series of locally integrable periodic real-valued functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Riemann-Lebesgue lemma, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                convolution product of periodic functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Dirichlet theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Fejer theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Parseval theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Fourier transforms on $\mathrm{L}^1(\R^d)$ and $\mathrm{L}^2(R^d)$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Plancherel’s theorem
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-10">
                                        <h5 class="card-title markdown-heading" id="10">
                                                <a role="button" data-toggle="collapse" href="#collapse-10" aria-expanded="false" aria-controls="collapse-10">
                                                        Probability Theory
                                                </a>
                                                <a class="hover-link" href="#10">#</a>
                                        </h5>
                                        <div id="collapse-10" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-10" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-10-0">
                                        <h5 class="card-title markdown-heading" id="10-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-10-0" aria-expanded="false" aria-controls="collapse-10-0">
                                                        Definitions of a probability space
                                                </a>
                                                <a class="hover-link" href="#10-0">#</a>
                                        </h5>
                                        <div id="collapse-10-0" class="collapse" data-parent="#overview-accordion-10 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-10-0" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                probability measure, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                events, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                independent events, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                sigma-algebra, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                independent sigma-algebra, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $0$-$1$ law, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Borel-Cantelli lemma, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                conditional probability, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                law of total probability
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-10-1">
                                        <h5 class="card-title markdown-heading" id="10-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-10-1" aria-expanded="false" aria-controls="collapse-10-1">
                                                        Random variables and their laws
                                                </a>
                                                <a class="hover-link" href="#10-1">#</a>
                                        </h5>
                                        <div id="collapse-10-1" class="collapse" data-parent="#overview-accordion-10 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-10-1" class="accordion">
                                                                
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                absolute continuity of probability laws, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                probability density function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                law of joint probability, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                independence of random variables, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                mean and variance of a real-valued random variable, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                transfer theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                moments, 
                        </span>
                        
                        
                        
						
                        
                        
                        
						
                        
                        <span class="statement-done">
                                binomial law, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                geometric law, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Poisson law, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                uniform law, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                exponential law, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Gaussian law, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                characteristic function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                probability generating functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                applications of probability generating functions to sums of independent random variables
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-10-2">
                                        <h5 class="card-title markdown-heading" id="10-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-10-2" aria-expanded="false" aria-controls="collapse-10-2">
                                                        Convergence of series of random variables
                                                </a>
                                                <a class="hover-link" href="#10-2">#</a>
                                        </h5>
                                        <div id="collapse-10-2" class="collapse" data-parent="#overview-accordion-10 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-10-2" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                convergence in probability, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $\mathrm{L}^p$ convergence, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                almost surely convergence, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Markov inequality, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Tchebychev inequality, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Levy's theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                weak law of large numbers, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                strong law of large numbers, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                central limit theorem
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-11">
                                        <h5 class="card-title markdown-heading" id="11">
                                                <a role="button" data-toggle="collapse" href="#collapse-11" aria-expanded="false" aria-controls="collapse-11">
                                                        Distribution calculus
                                                </a>
                                                <a class="hover-link" href="#11">#</a>
                                        </h5>
                                        <div id="collapse-11" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-11" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-11-0">
                                        <h5 class="card-title markdown-heading" id="11-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-11-0" aria-expanded="false" aria-controls="collapse-11-0">
                                                        Spaces $\mathcal{D}(\R^d)$
                                                </a>
                                                <a class="hover-link" href="#11-0">#</a>
                                        </h5>
                                        <div id="collapse-11-0" class="collapse" data-parent="#overview-accordion-11 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-11-0" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                smooth functions with compact support on $\R^d$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                stability by derivation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                stability by multiplication by a smooth function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                partitions of unity, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                constructing approximations of probability density functions in spaces of common functions (trig, exp, rational, log, etc)
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-11-1">
                                        <h5 class="card-title markdown-heading" id="11-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-11-1" aria-expanded="false" aria-controls="collapse-11-1">
                                                        Distributions on $\R^d$
                                                </a>
                                                <a class="hover-link" href="#11-1">#</a>
                                        </h5>
                                        <div id="collapse-11-1" class="collapse" data-parent="#overview-accordion-11 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-11-1" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                definition of distributions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                locally integrable functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                dirac measures, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Cauchy principal values, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                multiplication by a smooth function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                probability distribution function from a dataset, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                convergent distribution series, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                support of a distribution
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-11-2">
                                        <h5 class="card-title markdown-heading" id="11-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-11-2" aria-expanded="false" aria-controls="collapse-11-2">
                                                        Spaces $\mathcal{S}(\R^d)$
                                                </a>
                                                <a class="hover-link" href="#11-2">#</a>
                                        </h5>
                                        <div id="collapse-11-2" class="collapse" data-parent="#overview-accordion-11 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-11-2" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Schwartz space of rapidly decreasing functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                stability by derivation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                stability by multiplication by a slowly growing smooth function, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                gaussian functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Fourier transforms on $\mathcal{S}(\R^d)$, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                convolution of two functions of $\mathcal{S}(\R^d)$
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-11-3">
                                        <h5 class="card-title markdown-heading" id="11-3">
                                                <a role="button" data-toggle="collapse" href="#collapse-11-3" aria-expanded="false" aria-controls="collapse-11-3">
                                                        Tempered distributions
                                                </a>
                                                <a class="hover-link" href="#11-3">#</a>
                                        </h5>
                                        <div id="collapse-11-3" class="collapse" data-parent="#overview-accordion-11 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-11-3" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                definition, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                derivation of tempered distributions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                multiplication by a function $C^\infty$ of slow growth, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $L^2$ functions and Riesz representation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                $L^p$ functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                periodic functions, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Dirac comb, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Fourier transforms, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                inverse Fourier transform, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Fourier transform and derivation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Fourier transform and convolution product
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-11-4">
                                        <h5 class="card-title markdown-heading" id="11-4">
                                                <a role="button" data-toggle="collapse" href="#collapse-11-4" aria-expanded="false" aria-controls="collapse-11-4">
                                                        Applications
                                                </a>
                                                <a class="hover-link" href="#11-4">#</a>
                                        </h5>
                                        <div id="collapse-11-4" class="collapse" data-parent="#overview-accordion-11 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-11-4" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Poisson’s formula, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                using convolution and Fourier-Laplace transform to solve one dimensional linear differential equations, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                weak solution of partial derivative equation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                elementary solution of the Laplacian, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                solving the Laplace equations, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                heat equations, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                wave equations
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12">
                                        <h5 class="card-title markdown-heading" id="12">
                                                <a role="button" data-toggle="collapse" href="#collapse-12" aria-expanded="false" aria-controls="collapse-12">
                                                        Numerical Analysis
                                                </a>
                                                <a class="hover-link" href="#12">#</a>
                                        </h5>
                                        <div id="collapse-12" class="collapse" data-parent="#overview-accordion">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12" class="accordion">
                                                                
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12-0">
                                        <h5 class="card-title markdown-heading" id="12-0">
                                                <a role="button" data-toggle="collapse" href="#collapse-12-0" aria-expanded="false" aria-controls="collapse-12-0">
                                                        Solving systems of linear inequalities
                                                </a>
                                                <a class="hover-link" href="#12-0">#</a>
                                        </h5>
                                        <div id="collapse-12-0" class="collapse" data-parent="#overview-accordion-12 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12-0" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                conditioning, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Gershgorin-Hadamard theorem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Gauss’s pivot, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                LU decomposition
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12-1">
                                        <h5 class="card-title markdown-heading" id="12-1">
                                                <a role="button" data-toggle="collapse" href="#collapse-12-1" aria-expanded="false" aria-controls="collapse-12-1">
                                                        Iterative methods
                                                </a>
                                                <a class="hover-link" href="#12-1">#</a>
                                        </h5>
                                        <div id="collapse-12-1" class="collapse" data-parent="#overview-accordion-12 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12-1" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Jacobian, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Gauss-Seidel, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                convergence analysis, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                spectral ray, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                singular value decomposition, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                example of discretisation matrix by finite differences of the laplacian in one dimension
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12-2">
                                        <h5 class="card-title markdown-heading" id="12-2">
                                                <a role="button" data-toggle="collapse" href="#collapse-12-2" aria-expanded="false" aria-controls="collapse-12-2">
                                                        Iterative methods of solving systems of real and vector valued equations
                                                </a>
                                                <a class="hover-link" href="#12-2">#</a>
                                        </h5>
                                        <div id="collapse-12-2" class="collapse" data-parent="#overview-accordion-12 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12-2" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                linear systems case, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                proper element search, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                brute force method, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                optimization of convex function in finite dimension, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                gradient descent square root, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                nonlinear problems with real and vector values, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                bisection method, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Picard method, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Newton’s method, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                rate of convergence and estimation of error
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12-3">
                                        <h5 class="card-title markdown-heading" id="12-3">
                                                <a role="button" data-toggle="collapse" href="#collapse-12-3" aria-expanded="false" aria-controls="collapse-12-3">
                                                        Numerical integration
                                                </a>
                                                <a class="hover-link" href="#12-3">#</a>
                                        </h5>
                                        <div id="collapse-12-3" class="collapse" data-parent="#overview-accordion-12 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12-3" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Rectangle method, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                error estimation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Monte-Carlo method, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                rate of convergence, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                application to the calculation of multiple integrals
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12-4">
                                        <h5 class="card-title markdown-heading" id="12-4">
                                                <a role="button" data-toggle="collapse" href="#collapse-12-4" aria-expanded="false" aria-controls="collapse-12-4">
                                                        Approximation of numerical functions
                                                </a>
                                                <a class="hover-link" href="#12-4">#</a>
                                        </h5>
                                        <div id="collapse-12-4" class="collapse" data-parent="#overview-accordion-12 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12-4" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                Lagrange interpolation, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                Lagrange polynomial of a function at (n + 1) points, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                estimation of the error
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12-5">
                                        <h5 class="card-title markdown-heading" id="12-5">
                                                <a role="button" data-toggle="collapse" href="#collapse-12-5" aria-expanded="false" aria-controls="collapse-12-5">
                                                        Ordinary differential equations
                                                </a>
                                                <a class="hover-link" href="#12-5">#</a>
                                        </h5>
                                        <div id="collapse-12-5" class="collapse" data-parent="#overview-accordion-12 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12-5" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                numerical aspects of Cauchy's problem, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                explicit Euler method, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                consistency, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                stability, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                convergence, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                order
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
						
                        <div class="card statement-done">
                                <div class="card-header" id="heading-12-6">
                                        <h5 class="card-title markdown-heading" id="12-6">
                                                <a role="button" data-toggle="collapse" href="#collapse-12-6" aria-expanded="false" aria-controls="collapse-12-6">
                                                        Fourier transform
                                                </a>
                                                <a class="hover-link" href="#12-6">#</a>
                                        </h5>
                                        <div id="collapse-12-6" class="collapse" data-parent="#overview-accordion-12 ">
                                                <div class="card-body">
                                                        <div id="overview-accordion-12-6" class="accordion">
                                                                
						
                        
                        <span class="statement-done">
                                discrete Fourier transform on a finite abelian group, 
                        </span>
                        
                        
                        
						
                        
                        <span class="statement-done">
                                fast Fourier transform
                        </span>
                        
                        
                        
                                                        </div>
                                                </div>
                                        </div>
                                </div>
                        </div>
                        
                        
                                                        </div>
                                                </div>
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                        </div>
                        
                        
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